"Carrying On with Infinite Decimals" is an interesting read. As I was reading it, I thought about acyclic ("FIR" ?) _circuits_ for performing addition, which in real life are more-or-less asynchronous; a circuit element at depth N from a leaf has to wait until all of its inputs have stabilized before becoming stable itself. Within this circuit framework, different methods of carry propagation can be discussed, including the traditional right-to-left "ripple" carry method taught in grade school, and the "carry-lookahead" method taught in Digital Logic 101. Interestingly, I believe that the "carry lookahead" method solves the problem of these infinitely rippling carries. See also the recursive binary tree format for positional notation; a number representation is no longer a _sequence_ but a _tree_, where the sequence of leaves of the tree (the "fringe") is the traditional digit sequence. Vuillemin, Jean. "Efficient Data Structure and Algorithms for Sparse Integers, Sets and Predicates". IEEE Symp. on Computer Arith., Portland, OR, June, 2009. ISBN:: 978-0-7695-3670-5. http://www.ac.usc.es/arith19/sites/default/files/3670a007-session1-paper1.pd... When this binary tree representation is amended with the "quadtree hack", wherein any subtree all of whose digits are identical is replaced with that single digit, carry-lookahead addition becomes _the_ obvious/canonical method for addition. See the Lisp code at the following link for more details. http://home.pipeline.com/~hbaker1/rb.lsp --- Some infinite acyclic circuits can be modeled by circuits with cycles (e.g., "feedback", or the completely opaque "infinite impulse response"/IIR, circuits). Perhaps you can expand your paper along this line ? At 06:36 PM 11/18/2012, James Propp wrote:

If one is willing to live without subtraction, then there is a consistent mathematical structure in which .9999... is different from 1; see my preprint "Carrying On with Infinite Decimals" ( http://jamespropp.org/carrying.pdf), which my students presented at a recent Gathering for Gardner, and which I will eventually get around to publishing in one of the G4G books. (If any of the editors of the series are reading this, please feel free to gently hassle me about this, now or at some future time.)